Novel Criteria on Global Robust Exponential Stability to a Class of Reaction-Diffusion Neural Networks with Delays

Discrete Dynamics in Nature and Society Volume 2009, Article ID 291594,14pages doi:10.1155/2009/291594

Research Article

Novel Criteria on Global Robust Exponential Stability to a Class of Reaction-Diffusion Neural Networks with Delays

Jie Pan

^{1, 2}

and Shouming Zhong

¹

1College of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China

2Department of Applied Mathematics, Sichuan Agricultural University, Yaan, Sichuan 625014, China

Correspondence should be addressed to Jie Pan,[email protected] Received 13 June 2009; Accepted 31 August 2009

Recommended by Manuel De La Sen

The global exponential robust stability is investigated to a class of reaction-diffusion Cohen- Grossberg neural networkCGNNswith constant time-delays, this neural network contains time invariant uncertain parameters whose values are unknown but bounded in given compact sets.

By employing the Lyapunov-functional method, several new sufficient conditions are obtained to ensure the global exponential robust stability of equilibrium point for the reaction diffusion CGNN with delays. These sufficient conditions depend on the reaction-diffusion terms, which is a preeminent feature that distinguishes the present research from the previous research on delayed neural networks with reaction-diffusion. Two examples are given to show the effectiveness of the obtained results.

Copyrightq2009 J. Pan and S. Zhong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In recent years, considerable attention has been paid to study the dynamics of artificial neural networks with fixed parameters because of their potential applications in the areas such as signal and image processing, pattern recognition, parallel computations, and optimization problems1–10. However, during the implementation on very scale integration chips, the stability of a well-designed system may often be destroyed by its unavoidable uncertainty due to the existence of modelling error, external disturbance, and parameter fluctuation. In general, on other hand, a mathematical description is only an approximation of the actual physical system and deals with fixed nominal parameters. Usually, these parameters are not known exactly due to the imperfect identification or measurement, aging of components and/or changes in the environmental condition. Thus, it is almost impossible to get an exact model for the system due to the existence of various parameter uncertainties. So it is essential

to introduce the robust technique to design a system with such uncertainty11,12. If the uncertainty of a system is only due to the deviations and perturbations of its parameters, and if those deviations and perturbations are all bounded, then the system is called an interval system13–23. Recently, Chen and Rong23considered a class of Cohen-Grossberg neural networks CGNNs with time-varying delays. Several sufficient conditions were given to ensure global exponential robust stability.

In a real world, strictly speaking, the diffusion phenomena could not be ignored in neural networks and electric circuits once electrons transport in a nonuniform electromag- netic field. Hence, it is essential to consider the state variables varying with the time and space variables. The neural networks with diffusion terms can commonly be expressed by partial differential equations. Recently, some authors have devoted to the study of reaction- diffusion neural networks, for instance see24–29and references therein. In particular, more recently, Liu et al.27and Wang et al.28, considered the global exponential robust stability of a class of reaction-diffusion Hopfield neural networks with distributed delays and time- varying delay, respectively. Song and Cao29have obtained the criteria to guarantee the global exponential robust stability of a class of reaction-diffusion CGNNs with time-varying delays and Neumann boundary condition. In27–29, unfortunately, owing to the divergence theorem employed, a negative integral term with gradient is left out in their deduction. As a result, the global exponential robust stability criteria acquired by them do not contain a diffusion term. In other words, the diffusion term does not take effect in their deduction and sufficient conditions. The same case appears also in the other literatures24–26.

Motivated by the above discussions, in this paper we will consider a class of reaction- diffusion CGNNs with constant time delays and a boundary condition. We will construct a appropriate Lyapunov functional to derive some new criteria ensuring the global exponential robust stability for an equilibrium point of the delayed reaction-diffusion CGGNs with the boundary condition. The present work differs from the paper27–29sinceithe diffusion terms play an important role in the global exponential robust stability criteria in the paper, iithe boundary condition of CGNNs model considered includes the Neumann type and the Dirichlet type while the boundary condition of model in27,28is the Neumann type. The work will have significance impact on the design and applications of globally exponentially robustly stable reaction-diffusion neural network with delays and is of great interest in many applications.

The rest of this paper is organized as follows. In Section 2, model description and preliminaries are given. In Section3, several criteria are derived for the global exponential robust stability for an equilibrium point of reaction-diffusion CGNNs with delays and the boundary condition. Then, we give two examples and comparison to illustrate our criteria in Section4. Finally, in Section5, some conclusions are made.

2. Model Description and Preliminaries

To begin with, we introduce some notations.

i Ωis an open bounded domain inR^mwith smooth boundary∂Ω, and mesΩ>0 as mesΩdenotes the measure ofΩ.Ω Ω∪∂Ω.

iiL²Ωis the space of real Lebesgue measurable functions onΩwhich is a Banach space. Define the inner product u, v

Ωuvdx, for anyu, v ∈ L²Ω and the L²-normu2 : u, u^1/2, foru∈L²Ω.

iiiH¹Ω: {w ∈L²Ω, Diw ∈L²Ω}, whereDiw ∂w/∂xi, 1≤i≤ m.H₀¹Ω : the closure ofC₀^∞ΩinH¹Ω.

ivLetC: C0,∞×Ω,Rⁿbe the Banach space of continuous functions which map 0,∞×ΩintoRⁿwith the normu2 _n

i 1ui²₂^1/2foru u1, . . . , u_n^T ∈ C, whereui2

Ω|ui|²dx^1/2,i 1, . . . , n.

vLetC1 : C−τ,0×Ω,Rⁿbe the Banach space of bounded continuous functions which map−τ,0×ΩintoRⁿwith the following norm:φτ : sup_s∈−τ,0φs2, for anyφs φ1s, . . . , φns^T ∈ C1, whereφs_{C1 n}

i 1

Ω|φis|²dx^1/2. Consider the following reaction-diffusion CGNNs with interval coefficients and delays onΩ:

∂u_it, x

∂t

m l 1

∂

∂x_l

d_il∂u_it, x

∂x_l

−a_iuit, x

×

⎡

⎣b_iuit, x−ⁿ

j 1

s_ijf_j u_jt, x

−ⁿ

j 1

t_ijg_j g_j t−τ_ij, x Ii

⎤

⎦, t, x∈0,∞×Ω, 2.1

for i 1, . . . , n, x x1, . . . , x_m^T ∈ Ω is space variable, u_it, x corresponds to the state of the ith unit at time t and in space x; dil > 0, for i 1, . . . , n, l 1, . . . , m, corresponds to the transmission diffusion coefficient along theith neuron,d_i min_1≤l≤m{dil} fori 1, . . . , n;a_iuit, xrepresents an amplification function;b_iuit, xis an appropriate behavior function;sij,tijdenote the connection strengths of thejth neuron on theith neuron, respectively;g_jujt, x,f_jujt, xdenote the activation functions ofjth neuron at timet and in spacex;τ_ij0 < τ_ij ≤τcorresponds to the transmission delay along the axon of the jth unit from theith unit.Iiis the constant input from outside of the network.

Throughout this paper, we assume the following.

H1Each function a_iξ is positive, continuous, and bounded, that is, there exist constantsa_i,aisuch that 0< a_i≤aiξ≤ai<∞, forξ∈R,i 1, . . . , n.

H2Each functionbiξ∈ C¹R,Randb_iξ≥bi≥0 is locally Lipschitz continuous.

H3The activation functions f_jξ and g_jξsatisfy Lipschitz condition, that is, there exist two positive diagonal matricesF diagF1, . . . , F_nandG diagG1, . . . , G_n such that

fjξ1−fjξ2≤Fj|ξ1−ξ2|, gjξ1−gjξ2≤Gj|ξ1−ξ2|, 2.2

for allξ1, ξ2 ∈Rξ1/ξ2,j 1, . . . , n.

Remark 2.1. The activation functions f_j and g_j, j 1, . . . , n, are typically assumed to be sigmoidal which implies that they are monotone, bounded, and smooth. However, in this paper, we only need the previous weaker assumptions.

We assume that the nonlinear delayed systems 2.1 are supplemented with the boundary condition:

Buit, x 0, fort, x∈−τ,∞×∂Ω, i 1, . . . , n, 2.3

where Buit, x u_it, x is said the Dirichlet boundary condition, Buit, x

∂u_it, x/∂mis said the Neumann boundary condition, where∂u_it, x/∂m ∂uit, x/∂x1, . . .,∂uit, x/∂xm^T denotes the outward normal derivative on∂Ω.

Systems2.1are equipped with the initial condition:

u_is, x φ_is, x, fors, x∈−τ,0×Ω, i 1, . . . , n, 2.4

whereφ: φ1, . . . , φn^T ∈ C.

Given boundary condition 2.3 and initial function 2.4, the existence on the solutions of systems2.1, the reader can refer to18. We denote the solution byut, φ, x: u1t, φ, x,. . . , unt, φ, x^T and sometimes it is denoted byut, x,utorufor short when there is no risk of confusion.

Lemma 2.2. Under assumptions (H1)–(H3), system2.1has a unique equilibrium point, if H4b_i>Σⁿ_{j 1}s^∗_ijF_jt^∗_ijG_j, fori 1, . . . , n.

As for the proof of Lemma2.2, the reader can refer to21,28. Here, we omit it.

Definition 2.3. An equilibrium point u^∗ of system 2.1–2.4 is said to be globally exponentially stable onL²-norm, if there exist constantη >0 andM≥1 such that

ut, x−u^∗₂≤Mφ−u^∗

τe^−ηt ∀t≥0, 2.5

whereφ−u^∗τ sup_{−τ≤s≤0}φs, x−u^∗2.

Definition 2.4. Lets_ij ≤ s_ij ≤ s_ij, s^∗_ij max|s_ij|,|sij|, t_ij ≤ t_ij ≤ t_ij, t^∗_ij max|t_ij|,|tij|, I_i ≤ I_i ≤ I_i, I_i^∗ max|I_i|,|Ii|,τ_i max|τij|. An equilibrium pointu^∗ of system 2.1–

2.4is said to be globally exponentially robustly stable if its equilibrium pointu^∗is globally exponentially stable for alls_ij ≤ s_ij ≤ s_ij, t_ij ≤ t_ij ≤ t_ij, I_i ≤ I_i ≤ I_i, τ_ij ≤ τ_ij ≤ τ_ij, for i, j 1, . . . , n.

Lemma 2.5 Poincar´e inequality 30–32. Let Ω be a bounded domain of R^m with a smooth boundary∂Ωof classC²byΩ.vxis a real-valued function belonging toH₀¹ΩandBvx|_∂Ω 0. Then

λ1

Ω|vx|²dx≤

Ω|∇vx|²dx, 2.6

whichλ1is the lowest positive eigenvalue of the Laplacian with boundary condition

−Δψx λψx, x∈Ω, B

ψx

0, x∈∂Ω. 2.7

Regarding the proof of Lemma 2.5, we refer to any textbook on partial differential equations. For example,30,31or32are good standard references.

Remark 2.6. i When Ωis bounded or at least bounded in one direction, not only limited to a rectangle domain, inequality2.6holds. ii The lowest positive eigenvalueλ₁ of the Laplacian is sometimes known as the first eigenvalue. Determining the lowest eigenvalueλ1

is, in general, a very hard task that depends upon the geometry of the domain Ω. Certain special cases are tractable, however. For example, let the Laplacian onΩ {x1, x₂^T ∈R² | 0 < x1 < a, 0 < x2 < b}, if Bvx vx orBvx ∂vx/∂m, then λ1 π/a² π/b² orλ₁ min{π/a²,π/b²}, respectively.iii Although the eigenvalueλ₁ of the laplacian with the Dirichlet boundary condition on a generally bounded domainΩcannot be determined exactly, a lower bound of it may nevertheless be estimated byλ1 ≥m²/m 22π²/ω_m−11/V^2/m, whereω_m−1is a surface area of the unit ball inR^m,V is a volume of domainΩ 33.

3. Main Results

Theorem 3.1. Let hypotheses (H1)–(H4) hold. Assume further that A12diλ1a_ibi > ai_n

j 1s^∗_ijFi t^∗_ijGi _n

j 1ajs^∗_jiFj t^∗_jiGj,for i 1, . . . , n, then equilibrium pointu^∗of system2.1with2.3and2.4is globally exponentially robust stable for each constant inputI∈Rⁿ.

Proof. Lety_it u_it−u^∗.y_itis denoted byy_ifor short. From2.1, we obtain

∂yi

∂t m

l 1

∂

∂xl

dil

∂yi

∂xl

−aiui

⎡

⎣bi yi

−ⁿ

j 1

sijfj yj

−ⁿ

j 1

tijgj yj t−τij

⎤

⎦, 3.1

fort, x∈0,∞×Ω,i 1, . . . , n, where

b_i y_i

b_i y_iu^∗_i

−b_i u^∗_i

, f_j y_j f_j

y_ju^∗_j

−f_j u^∗_j

,

g_j y_j g_j

y_ju^∗_j

−g_j u^∗_j

,

3.2

fori, j 1, . . . , n.

Taking the inner product of both sides of3.1withyi, we get 1

2 d dty_i²

2

Ωy_i m

l 1

∂

∂x_l

d_il∂y_i

∂x_l

dx

Ωy_ia_iuiⁿ

j 1

s_ijf_j y_j dx

−

Ωy_ia_iuib_i y_i dx

Ωy_ia_iuiⁿ

j 1

t_ijg_j y_j t−τ_ij dx,

3.3

fort∈0,∞,i 1, . . . , n.

From the boundary condition2.3, Gauss formula and Lemma2.2, we have

Ωyi

m l 1

∂

∂x_l

dil

∂yi

∂x_l

dx −

Ω

m l 1

dil

∂yi

∂x_l 2

dx

≤ −di

Ω

m l 1

∂y_i

∂x_l 2

dx −di

Ω

∇yi²dx

≤ −λ1d_i

Ωy_i²dx −λ1d_iy_i²

2.

3.4

From assumptionH2, we get

Ωyiaiuibi yi

dx≥

Ωa_ibiyi²dx≥a_ibiyi²

2. 3.5

From assumptionsH1andH3, we obtain

Ωy_ia_iuiⁿ

j 1

s_ijf_j y_j

dx≤ⁿ

j 1

Ωs^∗_ija_iF_jy_iy_jdx≤ⁿ

j 1

s^∗_ija_iF_jy_i

2y_j

2. 3.6

By the same way, we have

Ωy_ia_iuiⁿ

j 1

t_ijg_j y_j t−τ_ij

dx≤ⁿ

j 1

t^∗_ija_iG_jy_i

2y_jt−τ_ij

2. 3.7

Combining3.4–3.7into3.3, we obtain d

dty_i²

2≤ −2 d_iλ₁a_ib_iy_i²

22 n j 1

s^∗_ija_iF_jy_i

2y_j

22 n

j 1

t^∗_ija_iG_jy_i

2y_jt−τ_ij

2, 3.8 fort∈0,∞.

According toA1, we can choose a sufficiently smallμ >0 such that

2 diλ1a_ibi

−μ−ⁿ

j 1

ai

s^∗_ijFit^∗_ijGi

−ⁿ

j 1

aj

s^∗_jiFjt^∗_jiGje^μτ

>0. 3.9

Now consider the Lyapunov functionalVtdefined by

Vt ⁿ

i 1

⎡

⎣y_i²

2e^μta_i n j 1

t^∗_ijG_{j t}

t−τij

e^μsτijy_js²

2ds

⎤

⎦. 3.10

By calculating the upper right Dini derivativeDVtofVtalong the solutions of3.1, we get

DVt e^μt n

i 1

⎧⎨

⎩μy_i²

2 d dty_i²

2a_i n

j 1

t^∗_ijG_je^μτijy_jt²

2

−a_i n j 1

t^∗_ijG_jy_jt−τ_ij²

2

⎫⎬

⎭

≤e^μt n

i 1

⎧⎨

⎩

−2 d_iλ₁a_ib_i

μy_i²

2

2 n j 1

s^∗_ijaiFjyi

2yj

22 n

j 1

t^∗_ijaiGjyi

2yjt−τij²

2

a_i n j 1

t^∗_ijG_je^μτijy_j²

2−a_i n j 1

t^∗_ijG_jy_jt−τ_ij²

2

⎫⎬

⎭

≤e^μt n

i 1

⎧⎨

⎩

−2 d_iλ₁a_ib_i

μy_i²

2

ⁿ

j 1

s^∗_ijaiFjyi²

2ⁿ

j 1

s^∗_ijaiFjyj²

2

ⁿ

j 1

t^∗_ijaiGjyi²

2ⁿ

j 1

t^∗_ijaiGjyjt−τij

2

a_i n j 1

t^∗_ijG_je^μτijy_j²

2−a_i n j 1

t^∗_ijG_jy_jt−τ_ij²

2

⎫⎬

⎭

≤e^μt n

i 1

⎧⎨

⎩

⎡

⎣−2 d_iλ₁a_ib_i

μa_i

⎛

⎝ⁿ

j 1

s^∗_ijF_jⁿ

j 1

t^∗_ijG_j

⎞

⎠

⎤

⎦y_i²

2

⎛

⎝ⁿ

j 1

s^∗_ija_iF_jⁿ

j 1

t^∗_ija_iG_je^μτij

⎞

⎠y_j²

2

⎫⎬

⎭

≤e^μt n

i 1

⎧⎨

⎩

⎡

⎣−2 diλ1a_ibi

μai

⎛

⎝ⁿ

j 1

s^∗_ijFjⁿ

j 1

t^∗_ijGj

⎞

⎠

⎤

⎦yi²

2

⎛

⎝ⁿ

j 1

s^∗_jiajFiⁿ

j 1

t^∗_jiajGie^μτ

⎞

⎠yi²

2

⎫⎬

⎭ e^μt

n i 1

⎧⎨

⎩−2 diλ1a_ibi

μai

n j 1

⎛

⎝s^∗_ijFjⁿ

j 1

t^∗_ijGj

⎞

⎠

ⁿ

j 1

aj

s^∗_jiFit^∗_jiGie^μτ⎫

⎬

⎭yi²

2,

3.11

fort∈0,∞. Hence

yt²

2e^μt≤Vt≤V0, fort∈0,∞. 3.12

Note that

V0 ⁿ

i 1

⎡

⎣y_i0²

2a_i n

j 1

t^∗_ijG_{j 0}

−τij

e^μsτijy_js²

2ds

⎤

⎦

≤

⎡

⎣1 1 μmax

1≤i≤n

⎧⎨

⎩ n j 1

a_jt^∗_jiG_iτ_jie^μτji

⎫⎬

⎭

⎤

⎦ⁿ

i 1

y_i²

τ.

3.13

DenoteM >0 and

M² 1 1 μmax

1≤i≤n

⎧⎨

⎩ n

j 1

a_jt^∗_jiG_iτ_jie^μτji

⎫⎬

⎭, 3.14

whereM >0, thenM≥1. So yt²

2≤M²φ−u^∗2

τe^−μt fort∈0,∞, 3.15

that is,

ut−u^∗₂≤Mφ−u^∗

τe^−1/2μt fort∈0,∞. 3.16

Since all the solutions of system2.1–2.4tend tou^∗exponentially ast → ∞for any values of the coefficients in system2.1with2.3-2.4, that is, the system described by2.1with 2.3-2.4has a unique equilibrium which is globally exponentially robust stable onL²-norm and the theorem is proved.

Remark 3.2. In the deduction for Theorem 3.1, by Lemma 2.5, we have obtained

−di

Ω|∇yi|²dx≤ −λ1diyit²₂see3.4. This is an important step. As a result, the condition of Theorem3.1includes the diffusion terms.

Changing a little the Lyapunov functional3.10by

Vt ⁿ

i 1

⎡

⎣yi²

2e^μtai

n j 1

tijG²_{j t}

t−τij

e^μsτijyjs²

2ds

⎤

⎦, 3.17

and using the similar way of the proof of Theorem3.1, we derive another new criterion.

Theorem 3.3. Under assumptions (H1)–(H4), if, in addition A22diλ₁a_ib_i>_n

j 1a_is^∗_ijt^∗_ijn

j 1a_js^∗_jiF_j²t^∗_jiG²_j, fori 1, . . . , n, then equilibrium pointu^∗ of system2.1 with2.3-2.4is globally exponentially robust stable for each constant inputI∈Rⁿ.

For system2.1, when the strength of the neuron interconnections sij and tij i, j 1, . . . , nis fixed constant matrices, the following result is obvious from Theorems3.1and3.3.

Corollary 3.4. Under assumptions (H1)–(H4), if any one of the following condition is true:

A32diλ1a_ibi> ai_n

j 1sijFitijGi _n

j 1ajsjiFjtjiGj, A42diλ₁a_ib_i> a_in

j 1sijtij_n

j 1a_jsjiF_j²tjiG²_j, fori 1, . . . , n, then equilibrium pointu^∗of system2.1with2.3-2.4is globally exponentially stable.

Remark 3.5. When a_iuit, x 1, b_iuit, x b_iu_it, x, i 1, . . . , n, then system 2.1 reduces to the following reaction-diffusion cellular neural network:

∂uit, x

∂t

m l 1

∂

∂x_l

dil∂uit, x

∂x_l

−

⎡

⎣biuit, x−ⁿ

j 1

sijfj ujt, x

−ⁿ

j 1

tijgj gj t−τij, x Ii

⎤

⎦, t, x∈0,∞×Ω, 3.18 fori 1, . . . , n.

From Theorems3.1and3.3, we have the following results.

Corollary 3.6. Under assumptions (H3) and (H4), if, in addition, any one of the following condition is true:

A32diλ1bi>_n

j 1s^∗_ijFit^∗_ijGi _n

j 1s^∗_jiFjt^∗_jiGj, A42diλ₁b_i > _n

j 1s^∗_ijt^∗_{ij n}

j 1s^∗_jiF_j² t^∗_jiG²_j, fori 1, . . . , n, then equilibrium pointu^∗of system 3.18with2.3and2.4is globally exponentially robust stable for each constant inputI ∈Rⁿ.

Remark 3.7. Whend_il 0, then system2.1reduces to the following system without diffusive terms:

∂u_it

∂t −aiuit

⎡

⎣b_iuit−ⁿ

j 1

s_ijf_j u_jtⁿ

j 1

t_ijg_j g_j t−τ_ij I_i

⎤

⎦, 3.19

fort≥0,i 1, . . . , n.

From Theorems3.1and3.3, we have the following results.

Corollary 3.8. Under assumptions (H1)–(H4), if, in addition, any one of the following condition holds:

A52a_ibi>_n

j 1aisijFitijGi _n

j 1ajsjiFjtjiGj, A62a_ibi>_n

j 1aisijtij _n

j 1ajsjiF_j²tjiG²_j, fori 1, . . . , n, then equilibrium point u^∗of system3.19with2.4is globally exponentially robust stable for each constant input I ∈Rⁿ.

Remark 3.9. From Theorems3.1and3.3, Corollary3.8, we see that conditionA5or condition A6implyA1andA2, respectively, conversely, if conditionsA1andA2hold,A5 and A6 do not certainly hold. This show that the reaction-diffusion terms have play an important role in the globally exponentially robust stability to a reaction-diffusion neural network.

4. Examples and Comparison

In order to illustrate the feasibility of the previous established criteria in the preceding sections, we provide concrete two examples. Although the selection of the coefficients and functions in the examples is somewhat artificial, the possible application of our theoretical theory is clearly expressed.

Example 4.1. Consider the following reaction-diffusion CGNNs onΩ {x x1, x2, x3^T ∈ R³|x²₁x²₂x²₃<1}:

∂

∂t u₁t

u₂t

⎡

⎢⎢

⎢⎣

0.65∂u₁t

∂x1 0.72∂u₁t

∂x2 0.65∂u₁t

∂x3

0.82∂u₂t

∂x₁ 0.65∂u₂t

∂x₂ 0.71∂u₂t

∂x₃

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

∂

∂x1

∂

∂x₂

∂

∂x3

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

−

10.2 cosu1t, x 0

0 10.2 sinu2t, x

×

#I1

I₂

1.2 0 0 1.2

u1t u₂t −

s11 s12

s₂₁ s₂₂

sinu1t cosu₂t

−

t₁₁ t₁₂ t₂₁ t₂₂

tanh u₁ t−τ_1j tanh u₂ t−τ_2j

$

, t, x∈0,∞×Ω,

u_it 0, t, x∈0,∞×∂Ω, i 1,2, u_is φ_is, s, x∈−1,0×Ω, i 1,2,

4.1

where tanhx e^x−e^−x/e^xe^−x,s₁₁∈1/5,1/4,s₁₂∈1/20,1/16,s₂₁∈−1/4,1/17, s22 ∈−1/4,−1/12,t11 ∈ 1/6,1/4,t12 ∈1/20,1/6,t21 ∈−1/4,−1/12,t22 ∈ 1/3,1/2, I1∈−1/5,1,I1∈−1/5,3/5,τ11∈0.3,0.8,τ12∈0.4,1,τ21 ∈0.1,0.6, andτ22 ∈0.2,0.9.

This model satisfies assumptionsH1–H4in this paper withλ₁ ≥0.5387,d₁ d₂ 0.65,a1 a2 1.2, a₁ a₂ 0.8,b1 b2 1.2,F1 F2 G1 G2 1, S^∗ s^∗_ijn×n

%_{1/4 1/16}

1/4 1/4

&

,T^∗ t^∗_ijn×n %_{1/4 1/16}

1/4 1/2

&

,τ 1,I₁^∗ 1,I₂^∗ 3/5. It is easily computed that

2.6204 2.6204

$

2 d_iλ₁a_ib_i

> a_i n j 1

s^∗_ijF_it^∗_ijG_i ⁿ

j 1

a_j

s^∗_jiF_jt^∗_jiG_j ⎧

⎨

⎩

1.9500, i 1, 2.5500, i 2.

4.2 From Theorem 3.1, we know that model 4.1 has a unique equilibrium point which is globally exponentially robustly stable.

Remark 4.2. It should be noted that

2.4 2a₂b2 ≯a2

n j 1

s^∗_2jF2t^∗_2jG2

ⁿ

j 1

aj

s^∗_j2Fjt^∗_j2Gj

2.5500. 4.3

From Corollary3.8, the corresponding delayed differential equation of system4.1without reaction-diffusion terms is not certainly robustly stable, as we can see in Example 4.1, reaction-diffusion terms do contribute to the exponentially robust stability of system4.1.

Example 4.3. For the model in Example4.1, if the diffusion operator,Ω, and the boundary condition are replaced by, respectively,

⎡

⎢⎢

⎢⎣

2∂u1t

∂x₁ 1.2∂u1t

∂x₂ 1.2∂u₂t

∂x1

2∂u₂t

∂x2

⎤

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎣

∂

∂x₁

∂

∂x2

⎤

⎥⎥

⎥⎦, Ω '

x1, x2^T ∈R²|0< xi< π, i 1,2(

, 4.4

and the Neumann boundary condition

∂uit

∂m 0, t, x∈0,∞×∂Ω, i 1,2, 4.5

the remainder parameters unchanged. According to Remark 2.1, we see that λ₁ 2. By Theorem3.1, using the same way with Example 4.1, we see that model4.1has a unique equilibrium point which is globally exponentially robustly stable.

Remark 4.4. Song and Cao have considered reaction-diffusion CGNNs with the Neumann boundary condition and obtained the criteria of the globally exponentially robust stability unique equilibrium point for CGNN, that is,29, Theorem 1. We notice that29, Theorem 1 is irrelevant to the reaction-diffusion terms. In principal, 29, Theorem 1 could be applied to analyze the globally exponentially robust stability for the system in Example 4.2.

Unfortunately,29, Theorem 1is not applicable to ascertain the globally exponentially robust stability for the system in Example 4.2, sinceaccording to the symbols in this paper

⎡

⎢⎢

⎢⎣ a₁b1

a1

0

0 a₂b2

a1

⎤

⎥⎥

⎥⎦−S^∗F−T^∗G

⎡

⎢⎢

⎣ 1 6 −1

8

−1 2 − 1

12

⎤

⎥⎥

⎦ 4.6

is not anM-matrix, whereF diag{F1, F₂},G diag{G1, G₂}.

5. Conclusion

In this paper, we have proposed several sufficient condition for the globally exponentially robustly stability of equilibrium point for the reaction-diffusion CGNNs with constant time delays. All the criteria are established by constructing suitable Lyapunov functionals, without assuming the monotonicity and differentiability of activation functions and the symmetry of connection matrices. The space domain that CGNNs model is on is relatively general, the boundary condition of CGNNs model includes the Dirichlet and the Neumann. In particular, Poincar´e inequality is used and all the criteria obtained depend on reaction-diffusion terms, this is a preeminent feature that distinguishes our research from the previous research on delayed neural network with reaction diffusion. Numerical examples are presented to illustrate the feasibility of this method.

Acknowledgment

The authors would like to thank the editor and the reviewers for their detailed comments and valuable suggestions which have led to a much improved paper.

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